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Properties

Label 1-571-571.149-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.655 - 0.755i$
Analytic cond. $2.65171$
Root an. cond. $2.65171$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.245 − 0.969i)2-s + (−0.909 − 0.416i)3-s + (−0.879 − 0.475i)4-s + (0.701 − 0.712i)5-s + (−0.627 + 0.778i)6-s + (−0.340 + 0.940i)7-s + (−0.677 + 0.735i)8-s + (0.652 + 0.757i)9-s + (−0.518 − 0.854i)10-s + (−0.518 + 0.854i)11-s + (0.601 + 0.799i)12-s + (0.863 + 0.504i)13-s + (0.828 + 0.560i)14-s + (−0.934 + 0.355i)15-s + (0.546 + 0.837i)16-s + (0.0495 − 0.998i)17-s + ⋯
L(s)  = 1  + (0.245 − 0.969i)2-s + (−0.909 − 0.416i)3-s + (−0.879 − 0.475i)4-s + (0.701 − 0.712i)5-s + (−0.627 + 0.778i)6-s + (−0.340 + 0.940i)7-s + (−0.677 + 0.735i)8-s + (0.652 + 0.757i)9-s + (−0.518 − 0.854i)10-s + (−0.518 + 0.854i)11-s + (0.601 + 0.799i)12-s + (0.863 + 0.504i)13-s + (0.828 + 0.560i)14-s + (−0.934 + 0.355i)15-s + (0.546 + 0.837i)16-s + (0.0495 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.655 - 0.755i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.655 - 0.755i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4380490056 - 0.9605760080i\)
\(L(\frac12)\) \(\approx\) \(0.4380490056 - 0.9605760080i\)
\(L(1)\) \(\approx\) \(0.6962415725 - 0.5845763817i\)
\(L(1)\) \(\approx\) \(0.6962415725 - 0.5845763817i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.245 - 0.969i)T \)
3 \( 1 + (-0.909 - 0.416i)T \)
5 \( 1 + (0.701 - 0.712i)T \)
7 \( 1 + (-0.340 + 0.940i)T \)
11 \( 1 + (-0.518 + 0.854i)T \)
13 \( 1 + (0.863 + 0.504i)T \)
17 \( 1 + (0.0495 - 0.998i)T \)
19 \( 1 + (0.115 - 0.993i)T \)
23 \( 1 + (0.490 - 0.871i)T \)
29 \( 1 + (-0.401 - 0.915i)T \)
31 \( 1 + (0.546 + 0.837i)T \)
37 \( 1 + (-0.213 - 0.976i)T \)
41 \( 1 + (-0.986 + 0.164i)T \)
43 \( 1 + (0.922 + 0.386i)T \)
47 \( 1 + (0.546 + 0.837i)T \)
53 \( 1 + (0.371 - 0.928i)T \)
59 \( 1 + (0.789 + 0.614i)T \)
61 \( 1 + (0.997 - 0.0660i)T \)
67 \( 1 + (0.371 - 0.928i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.213 - 0.976i)T \)
79 \( 1 + (-0.724 - 0.689i)T \)
83 \( 1 + (-0.627 + 0.778i)T \)
89 \( 1 + (-0.627 + 0.778i)T \)
97 \( 1 + (0.991 - 0.131i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.50937654423083366853533258804, −22.805974271285191880321674674341, −22.13275779521066345747597050935, −21.371961808322608588849604004558, −20.62381150910050572359179671229, −18.85581477841872112100115098162, −18.40707429648599705679047762423, −17.22902595399650941234415648606, −17.03417321759354134169560625388, −15.998276238439799819200939255382, −15.30939416569483883036846761096, −14.30055245175956449730128748043, −13.39654642929020902559260597056, −12.88578858185612586523176231735, −11.43124167543511870538846302253, −10.43306355373758150747197880445, −10.02524034098965173830691518745, −8.72423543167403348521202073620, −7.569806178238511404481861871513, −6.64318284655248647773280373898, −5.89602625822625749389909100210, −5.35977211963181397790960047661, −3.86503834072710314659611147626, −3.361702957290672875070305502957, −1.09873248656919047607391677038, 0.7033443067332565109435970085, 1.92890482727146931822754142232, 2.65764370188565941660889042988, 4.436988356888025194151809849397, 5.11777059109507809509522743831, 5.863531845199553188356836765498, 6.87749061284278922980709754102, 8.48142919310161476437892638086, 9.316965995472865726267263685753, 10.0746365425843886282509888708, 11.13752290889049868949279105627, 11.91498115577573619616565483551, 12.687399014775164675141177601974, 13.19036471688372637272467705015, 14.063207950520113243058247952504, 15.48649439849238521861127811815, 16.26386315406353939683668671605, 17.45918362593320593233921043479, 18.038105651286652168665491342463, 18.66705969460143267227186765539, 19.57247139232172660236787970636, 20.81338095915255251273157762484, 21.14230622840118533106457503317, 22.1775920230273209231428328036, 22.74480561117240690994406244309

Graph of the $Z$-function along the critical line